Number Theory: Patterns, Primes and Proofs

Bring with you: Read everyday English, count objects, and add, subtract, multiply and divide small positive whole numbers. A calculator is welcome.

See why number patterns work, solve grouping and timing puzzles, and learn the difference between a guess and a proof.

A beginner course with explicit foundations and plenty of practice. Begin with number families, signs, fractions and decimals. Then study divisibility, primes, common divisors, remainders and simple proofs. No algebra or proof experience is assumed. Allow about 7 to 8 hours, including written reasoning and retries.

By the end, you can…

• Explain counting numbers, whole numbers, integers, rational and irrational numbers, and how they fit on the real number line.
• Use exact grouping and remainders, distinguish odd from even, and identify primes with reasons.
• Factor numbers, interpret powers, find a greatest common divisor and explain the Euclidean algorithm.
• Find a least common multiple, calculate with remainders and use repeating powers to solve digit problems.
• Solve small remainder puzzles, coordinate two cycles, explain why primes never end and combine these tools independently.

For another Course

• A full university number-theory syllabus, including quadratic reciprocity, arithmetic functions and analytic number theory.
• The general proofs of unique prime factorization, modular-inverse existence and the Chinese remainder theorem.
• Cryptographic implementation, Fermat's little theorem, advanced Diophantine equations and continued fractions.
• Full arithmetic with general fractions and real numbers, percentages, complex-number calculations, and proofs that π or √2 are irrational; the required meanings and examples are included.
• Special divisibility shortcuts for 7 and 11, perfect/abundant/deficient numbers, advanced divisor functions and continued-fraction approximations. These are extensions, not prerequisites for the stated beginner outcomes.